For a positive integer $n$ let
$$\begin{eqnarray}\mathfrak{P}_{n}=\mathop{\prod }_{\substack{ p \\ s_{p}(n)\geqslant p}}p,\end{eqnarray}$$ where
$p$ runs over primes and
$s_{p}(n)$ is the sum of the base
$p$ digits of
$n$. For all
$n$ we prove that
$\mathfrak{P}_{n}$ is divisible by all “small” primes with at most one exception. We also show that
$\mathfrak{P}_{n}$ is large and has many prime factors exceeding
$\sqrt{n}$, with the largest one exceeding
$n^{20/37}$. We establish Kellner’s conjecture that the number of prime factors exceeding
$\sqrt{n}$ grows asymptotically as
$\unicode[STIX]{x1D705}\sqrt{n}/\text{log}\,n$ for some constant
$\unicode[STIX]{x1D705}$ with
$\unicode[STIX]{x1D705}=2$. Further, we compare the sizes of
$\mathfrak{P}_{n}$ and
$\mathfrak{P}_{n+1}$, leading to the somewhat surprising conclusion that although
$\mathfrak{P}_{n}$ tends to infinity with
$n$, the inequality
$\mathfrak{P}_{n}>\mathfrak{P}_{n+1}$ is more frequent than its reverse.